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Lecture Summaries for Differential Equations
Lecture 01: An introduction to the very basic
definitions and terminology of differential equations,
as well as a discussion of central issues and objectives for the course.
Lecture 02: Solving first order linear differential equations and initial
value problems using
integrating factors .
Lecture 03: Solving separable equations.
Lecture 04: The Existence and Uniqueness Theorem for solving general
first order linear equations.
Lecture 05: Applications of first order ODEs involving continuous
compounding, and population
dynamics using the logistic equation .
Lecture 06: Solving the logistic equation, and an application of first
order ODEs to a problem
of physics.
Lecture 07: Solving exact equations.
Lecture 08: Sketching a proof of the Existence and Uniqueness Theorem for
first order ODEs.
Lecture 09: An introduction to difference equations and their solutions,
focusing on first order
linear difference equations.
Lecture 10: An application of first order linear difference equations, as
well as a brief discussion
of non linear difference equations, their solutions, and stairstep diagrams.
Lecture 11: An introduction to second order ODEs and initial value
problems, and a discussion
of solutions to second order homogeneous constant coefficient equations.
Lecture 12: A discussion of existence and uniqueness results for second
order linear ODEs, and
of fundamental sets of solutions and the importance of the Wronskian of
solutions.
Lecture 13: A discussion of the structure of the set of solutions to a
linear homogeneous ODE
from a linear algebra perspective ; concepts such as linear independence, span,
and basis are used
to better understand fundamental sets of solutions .
Lecture 14: Solving ODEs with characteristic equation having nonreal
complex roots.
Lecture 15: Solving ODEs with characteristic equation having repeated
roots.
Lecture 16: Solving second order linear nonhomogeneous equations using
the method of undetermined
coefficients .
Lecture 17: Solving second order linear nonhomogeneous equations using
the method of variation
of parameters.
Lecture 18: A discussion of the structure of solution sets to higher
order linear equations, the
basic Existence and Uniqueness Theorem, and a generalization of the Wronskian.
Lecture 19: Solving higher order constant
coefficient homogeneous equations.
Lecture 20: Solving higher order nonhomogeneous equations using the
method of undetermined
coefficients .
Lecture 21: Solving higher order nonhomogeneous equations using the
method of variation of
parameters.
Lecture 22: A review of the most fundamental properties of power series.
Lecture 23: Solving differential equations and initial value problems
using power series.
Lecture 24: An example of how to use power series to solve nonconstant
coefficient ODEs,
and a discussion of the basic theorem underlying the use of power series to
solve ODEs.
Lecture 25: A review of improper integration and an introduction to the
Laplace transform.
Lecture 26: A discussion of the main properties of the Laplace transform
which make it useful
for solving initial value problems.
Lecture 27: A discussion of how the Laplace transform and its inverse act
on unit step functions,
exponentials , and products of these functions with others.
Lecture 28: An introduction to the convolution of two functions, and an
examination of how
the Laplace transform acts on such a convolution.
Lecture 29: An introduction to systems of equations and the basic
existence and uniqueness
result for the corresponding initial value problems .
Lecture 30: An introduction to vector function notation, and a discussion
of the structure of
solution sets to homogeneous systems and the importance of the Wronskian.
Lecture 31: Solving constant coefficient linear homogeneous systems using
eigenvalues and
eigenvectors.
Lecture 32: Solving constant coefficient linear homogeneous systems in
the case where an
eigenvalue is complex .
Lecture 33: Solving constant coefficient linear homogeneous systems in
the case where there is
a repeated eigenvalue.
Lecture 34: Viewing solutions to linear homogeneous systems in terms of
fundamental matrices
and the exponential of a matrix .
Lecture 35: Solving nonhomogeneous systems using diagonalization and
variation of parameters.
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